Drawing a Triangle on the Thurston Model of Hyperbolic Space

Summary: This article carefully compares the classical Poincaré disk model of the hyperbolic plane with a physical paper model due to Thurston. The authors present a clear discussion of the differences between the models, including a comparison of lines in the Thurston model with true hyperbolic lines. They include new insights into a classical theorem from differential geometry.

Curtis Bennett earned a B.S. in mathematics from Colorado State University in 1985 and received his Ph.D. in mathematics from the University of Chicago in 1990. Since graduating, he has taught at Michigan State University, Ohio State University, Bowling Green State University, and Loyola Marymount University. In 1993, Dr. Bennett was a co-founder of the Young Mathematicians Network and served on the editorial board of the YMN for three years. He works in a variety of fields, including the study of geometries associated to groups of Lie type, combinatorics, and the scholarship of teaching and learning. He was a 2000-2001 and 2003-2004 Carnegie Scholar with the Carnegie Foundation for the Advancement of Teaching, and he won the Haimo Award for Excellence in Teaching in 2010. In his spare time, he enjoys playing golf (badly), bicycling, and hiking.

Blake Mellor earned a B.A. in mathematics from Harvard University in 1993 and received his Ph.D. in mathematics from UC Berkeley in 1999, under the direction of Robion Kirby. After three years at Florida Atlantic University, he moved to Loyola Marymount University in Los Angeles, where he is currently an associate professor of mathematics. His research interests are in knot theory and spatial graphs, with occasional forays exploring connections between mathematics and the arts. He enjoys martial arts, ballroom dancing, science fiction, and playing with his children, Eric and Clara.

Patrick D. Shanahan earned a B.A. in mathematics from California State University, Long Beach in 1990. He attended graduate school at UC Santa Barbara where he completed an M.A. in 1992 and a Ph.D. in 1996, supervised by Daryl Cooper. A highlight in his graduate career was participating in the MSRI summer graduate student program in hyperbolic geometry led by David Epstein, Jane Gilman, and Bill Thurston. He joined the faculty at Loyola Marymount University in 1996 where he is currently a professor of mathematics. Professor Shanahan's main area of research is in geometric topology with an emphasis on knot theory. He has also co-authored the textbook A First Course in Complex Analysis with Applications, currently in its second edition. In his spare time you can find him at the beach surfing with his children, Kasey and Cody.

This article carefully compares the classical Poincaré disk model of the hyperbolic plane with a physical paper model due to Thurston. The authors present a clear discussion of the differences between the models, including a comparison of lines in the Thurston model with true hyperbolic lines. They include new insights into a classical theorem from differential geometry.